On normalizations of Thurston measure on the space of measured laminations

Monin, Leonid; Telpukhovskiy, Vanya

doi:10.1016/j.topol.2019.106878

Cited by 14 publications

(12 citation statements)

References 8 publications

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“…of the counting measure on integral multi-curves. For a discussion of how the measure defined by the symplectic structure (which is what we use in this paper) is related to the measure defined by the scaling limit (used by Mirzakhani in [15]) see [16] The main idea in [15] is to relate the counting problem in (1.1) to the Weil-Petersson volume of certain sets in T . More precisely, for a filling curve γ and L > 0, Mirzakhani considered the sets (Papadopoulos).…”

Section: Weil-petersson Volumementioning

confidence: 99%

Geodesic Currents and Counting Problems

Rafi

Souto

2019

Geom. Funct. Anal.

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Section: Weil-petersson Volumementioning

confidence: 99%

Geodesic Currents and Counting Problems

Rafi

Souto

2019

Geom. Funct. Anal.

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“…where the sum is taken over all measured laminations γ ∈ ML 0 (S) corresponding to simple multi-curves with integral weights. In fact, the ratio between m Th and m sympl Th is a constant factor that only depends on the topology of S (see [19], for instance). In what follows, we refer to m Th as the Thurston measure.…”

Section: Thurston Measure and Ergodicitymentioning

confidence: 99%

Ergodic invariant measures on the space of geodesic currents

Erlandsson¹,

Mondello

2022

Annales De l'Institut Fourier

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Let S be a compact, connected, oriented surface, possibly with boundary, of negative Euler characteristic. In this article we extend Lindenstrauss-Mirzakhani's and Hamenstädt's classification of locally finite mapping class group invariant ergodic measures on the space of measured laminations ML (S) to the space of geodesic currents C (S), and we discuss the homogeneous case. Moreover, we extend Lindenstrauss-Mirzakhani's classification of orbit closures to C (S). Our argument relies on their results and on the decomposition of a current into a sum of three currents with isotopically disjoint supports: a measured lamination without closed leaves, a simple multi-curve and a current that binds its hull.Résumé. -Soit S une surface compacte, connexe, orientée, éventuellement à bord, de caractéristique d'Euler négative. Dans cet article nous étendons la classification des mesures ergodiques, localement finies et invariantes sous l'action du mapping class group, sur l'espace des laminations mesurées ML (S) obtenue par Lindenstrauss-Mirzakhani et Hamenstädt, à l'espace des courants géodésiques C (S), et nous discutons le cas homogène. De plus, nous étendons la classification de la fermeture des orbites obtenue par Lindenstrauss-Mirzakhani à C (S). Notre argument repose sur leurs résultats et sur le décomposition d'un courant en une somme de trois courants avec supports isotopiquement disjoints: une lamnation mesurée sans feuilles fermées, une multi-courbe simple et un courant qui remplit son enveloppe. Viveka ERLANDSSON & Gabriele MONDELLO SettingLet S be a smooth, compact, connected, oriented surface of negative Euler characteristic, possibly with boundary, and let Map(S) be its mapping class group, i.e. the group of isotopy classes of orientation-preserving diffeomorphisms S → S that send each boundary curve of S to itself.Consider an auxiliary hyperbolic metric on S such that ∂S is geodesic. A geodesic current on S is a π 1 (S)-invariant Radon measure on the space G( S) of bi-infinite geodesics in the universal cover S of S. The space C (S) of all geodesic currents, naturally endowed with the weak -topology, can also be viewed as the completion of the set of weighted closed curves on S in the same way as the space ML (S) of measured laminations is the completion of the set of weighted simple closed curves. Recall that a measured lamination is a closed subset of S foliated by complete geodesics and endowed with a transverse measure of full support. Hence a measured lamination can be viewed as a current and ML (S) can be viewed as a subspace of C (S). The geometric intersection number of closed curves has a unique continuous extension to a symmetric, bi-homogenous intersection form). The subspace of measured laminations consists exactly of those currents c for which ι(c, c) = 0.The aim of this paper is to provide a classification of locally finite ergodic measures on C (S) that are invariant under the natural action of Map(S) and of closures of Map(S)-orbits on C (S). MotivationThe impetus for the present paper -in add...

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“…It follows that the measure induced by the Thurston volume form on ML g,n is a multiple of µ Thu . Moreover, see [MT19] for a detailed proof, the scaling factor relating these measures can be computed explicitely:…”

Section: Background Materialsmentioning

confidence: 99%

Square-Integrability of the Mirzakhani Function and Statistics of Simple Closed Geodesics on Hyperbolic Surfaces

Arana-Herrera

Athreya

2020

Forum of Mathematics, Sigma

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Given integers g, n ≥ 0 satisfying 2 − 2g − n < 0, let Mg,n be the moduli space of connected, oriented, complete, finite area hyperbolic surfaces of genus g with n cusps. We study the global behavior of the Mirzakhani function B : Mg,n → R ≥0 which assigns to X ∈ Mg,n the Thurston measure of the set of measured geodesic laminations on X of hyperbolic length ≤ 1. We improve bounds of Mirzakhani describing the behavior of this function near the cusp of Mg,n and deduce that B is square-integrable with respect to the Weil-Petersson volume form. We relate this knowledge of B to statistics of counting problems for simple closed hyperbolic geodesics. Proposition 1.2. [Mir08b, Proposition 3.2 and Theorem 3.3] The function B : M g,n → R >0 is continuous, proper, and integrable with respect to the Weil-Petersson volume form on M g,n .Let µ wp be the measure induced by the Weil-Petersson volume form on M g,n . We consider the normalization constant b g,n := Mg,n B(X) d µ wp (X) < ∞.The last result needed to describe the leading coefficient n γ (X) is the following proposition, corresponding to Corollary 5.2 in [Mir08b]. Proposition 1.3. [Mir08b, Corollary 5.2] For any rational multi-curve γ on S g,n , the integral P (L, γ) := Mg,n s(X, γ, L) d µ wp (X) is a polynomial of degree 6g − 6 + 2n in L > 0 with non-negative coefficients and whose leading coefficient c(γ) := lim L→∞ P (L, γ) L 6g−6+2n is a positive rational number.

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On normalizations of Thurston measure on the space of measured laminations

Cited by 14 publications

References 8 publications

Geodesic Currents and Counting Problems

Geodesic Currents and Counting Problems

Ergodic invariant measures on the space of geodesic currents

Square-Integrability of the Mirzakhani Function and Statistics of Simple Closed Geodesics on Hyperbolic Surfaces

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